ממן17

ממן17 - 17 " 1...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 17 " 1 . ( AB )v = v - v B 0, v F n . AB .1 , B - . A( Bv) = v , . ( BA)u = u : , u = Bv . ( BA)( Bv) = B [ A( Bv) ] = B ( v) = ( Bv) . BA - , u = Bv 0 0 . = 0 , v B 0 . v = ( AB )v = A( Bv ) = A = 0 , u = Bv = 0 - ( AB ) x = 0 , v B 0 - ( AB )v = 0 - , BA = B A = A B = AB = 0 , AB = 0 , AB , ( BA) x = 0 - , . BA . BA = 0 , AB BA BA - AB - . B - A . T , S : F 2 B F 2 : . T ( x, y ) := ( 1 3 y, 2 x ) S ( x, y ) := ( y - x, x + y ) : TS (1, 2) = T ( S (1, 2)) = T (1,3) = (1, 2) = 1 (1, 2) ST (1, 2) = S (T (1, 2)) = S ( 2 3 , 2) = ( 4 3 , 8 3 ) = 4 3 (1, 2) , = 1 TS v = (1, 2) , . ST .2 , T , S , , A, B : 0 - 13 1 1 A= , B = 1 2 1 0 : 1 2 3 1 3 - 13 AB = , BA = - 2 2 2 13 1 4 1 1 2 1 3 1 3 1 - 1 3 1 3 4 ( AB )v = = 1 , ( BA)v = = = 3 2 2 2 8 = - 2 2 2 2 2 13 3 , = 1 AB v = (1, 2) . BA 2 . v U , v = ( x, y , z ) U . U - , 3 x + 12 y - z = 0 , . ( x, y, z ) B(3,12, -1) = 0 , v (3,12, -1) ,VIII.7 ) v = ( x, y ,3 x + 12 y ) = x(1, 0,3) + y (0,1,12) = xu1 + yu2 , - u1 , u2 - . U = Sp{u1 , u2 } , .( - . dim U = 2 , U - B = (u1 , u2 ) : U - B * * u2 . u1 = u1 = (1, 0,3) * u2 Bu1 .1 * u 2 = u2 - u * 2 1 * u1 = (0,1,12) - 36 (1, 0,3) = ( - 18 5 ,1, 6 5 ) 10 * : , 5 - u2 * u2 = ( -18,5, 6) , U - 2 ^ : B , . U - u * u * 1 1 ^ B = 1 , 2 (1, 0,3), = (-18,5, 6) = * * u1 u2 385 10 1 18 3 5 6 = , , , , 0, - 10 385 385 385 10 . u1 = (3,12, -1) . U ((3,12, -1)) : , U v = ( -10, 2,8) u1 .2 [ u1 u1 ] [ ]= [v u1 ] : 154 = -14 U v , . - 111 . u = - 111 u1 = (- 311 , - 12 11 , 111) 3 :- v + w - v - w .- w - v .1 1 (v - w) + 2 (v + w) = 0 (1 + 2 )v + (2 - 1 ) w = 0 1 + 2 = 2 - 1 = 0 , . - v + w - v - w , 1 = 2 = 0 , , U = Sp{v - w, v + w} (v - w, v + w) (*) dim U = 2 U - ) Sp{u} U , u U - . dimSp{u} dim U , Sp{u} U , .( Sp{u} , dimSp{u} = 1 , Sp{u} - (u ) , u B 0 (**) dim U 1 (**)- (*)- dim U + dim U Rn = U U dim R n = dim U + dim U : dim R n B 3 , .n B3 w B Im T . Tu Bv = 0 v B ker T u B R n v B ker T w B Im T , . Tu = w - u B R n , v .2 3 , ( Im T ) , v Im T v B ker T , . w Bv = 0 . . ker T ( Im T ) . ker T = ( Im T ) , ( Im T ) ker T - , ( Im T ) = {0} , ( Im T ) = R n , Im T = {0} . v ( Im T ) , v 0 , ( Im T ) {0} , w B Im T . Im T B {0} - . ker T = ( Im T ) , ker T = R n . Tu Bv = 0 , Tu = w - u B R n . w Bv = 0 4 .1 5 . dimU 2 = n - k , , dimU1 = k . R n = U1 U 2 .1 , R n = U1 U1 = U 2 U 2 , , , . dim U1 = n - k , dim U 2 = k n (1) dim U1 + dim U 2 = dim R : U . u U 2 u U1 , , u U 2 u U1 . u 1 U 2 K1 = {u1 ,K , uk } U1 , K 2 = {v1 ,K vn - k } U 2 , K1 , K 2 u ,VIII.7 . U1 = Sp K1 , U 2 = Sp K 2 , R n = U1 + U 2 R n = U1 U 2 . K1 B K 2 K 2 ) : (2) U1 U 2 = {0} - , dim(U1 + U 2 ) = dim U1 + dim U 2 = dim R n ,(2)- (1) . u = 0 , . u R n ,VIII.7 , . R n = Sp( K1 . R n = U1 U 2 , U1 U 2 = {0} ,(2) . R n = U1 + U 2 , U1 + U 2 Rn - . U1 = U 2 = R n : . . Rn U1 U 2 - , U1 = U 2 = {0} , R n = U1 + U 2 + .2 ...
View Full Document

Ask a homework question - tutors are online